\(\int F^{c (a+b x)} \sinh ^3(d+e x) \, dx\) [15]

Optimal result

Integrand size = 18, antiderivative size = 132 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=-\frac {3 e^{-d-e x} F^{c (a+b x)}}{8 (e-b c \log (F))}+\frac {e^{-3 d-3 e x} F^{c (a+b x)}}{8 (3 e-b c \log (F))}-\frac {3 e^{d+e x} F^{c (a+b x)}}{8 (e+b c \log (F))}+\frac {e^{3 d+3 e x} F^{c (a+b x)}}{8 (3 e+b c \log (F))} \] Output:

-3*exp(-e*x-d)*F^(c*(b*x+a))/(8*e-8*b*c*ln(F))+exp(-3*e*x-3*d)*F^(c*(b*x+a 
))/(24*e-8*b*c*ln(F))-3*exp(e*x+d)*F^(c*(b*x+a))/(8*e+8*b*c*ln(F))+exp(3*e 
*x+3*d)*F^(c*(b*x+a))/(24*e+8*b*c*ln(F))
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.19 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\frac {F^{c (a+b x)} \left (3 \cosh (3 (d+e x)) \left (e^3-b^2 c^2 e \log ^2(F)\right )+3 \cosh (d+e x) \left (-9 e^3+b^2 c^2 e \log ^2(F)\right )+2 b c \log (F) \left (13 e^2-b^2 c^2 \log ^2(F)+\cosh (2 (d+e x)) \left (-e^2+b^2 c^2 \log ^2(F)\right )\right ) \sinh (d+e x)\right )}{4 \left (9 e^4-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)\right )} \] Input:

Integrate[F^(c*(a + b*x))*Sinh[d + e*x]^3,x]
 

Output:

(F^(c*(a + b*x))*(3*Cosh[3*(d + e*x)]*(e^3 - b^2*c^2*e*Log[F]^2) + 3*Cosh[ 
d + e*x]*(-9*e^3 + b^2*c^2*e*Log[F]^2) + 2*b*c*Log[F]*(13*e^2 - b^2*c^2*Lo 
g[F]^2 + Cosh[2*(d + e*x)]*(-e^2 + b^2*c^2*Log[F]^2))*Sinh[d + e*x]))/(4*( 
9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.44, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5999, 5997}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sinh ^3(d+e x) F^{c (a+b x)} \, dx\)

\(\Big \downarrow \) 5999

\(\displaystyle -\frac {6 e^2 \int F^{c (a+b x)} \sinh (d+e x)dx}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \sinh ^3(d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {3 e \sinh ^2(d+e x) \cosh (d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}\)

\(\Big \downarrow \) 5997

\(\displaystyle -\frac {b c \log (F) \sinh ^3(d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {3 e \sinh ^2(d+e x) \cosh (d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {6 e^2 \left (\frac {e \cosh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \sinh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}\right )}{9 e^2-b^2 c^2 \log ^2(F)}\)

Input:

Int[F^(c*(a + b*x))*Sinh[d + e*x]^3,x]
 

Output:

(3*e*F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^2)/(9*e^2 - b^2*c^2*Log[F 
]^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x]^3)/(9*e^2 - b^2*c^2*Log[F 
]^2) - (6*e^2*((e*F^(c*(a + b*x))*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2) 
- (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2)))/(9 
*e^2 - b^2*c^2*Log[F]^2)
 

Defintions of rubi rules used

rule 5997

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)], x_Symbol] : 
> Simp[(-b)*c*Log[F]*F^(c*(a + b*x))*(Sinh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2 
)), x] + Simp[e*F^(c*(a + b*x))*(Cosh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2)), x 
] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]
 

rule 5999

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)]^(n_), x_Symb 
ol] :> Simp[(-b)*c*Log[F]*F^(c*(a + b*x))*(Sinh[d + e*x]^n/(e^2*n^2 - b^2*c 
^2*Log[F]^2)), x] + (Simp[e*n*F^(c*(a + b*x))*Cosh[d + e*x]*(Sinh[d + e*x]^ 
(n - 1)/(e^2*n^2 - b^2*c^2*Log[F]^2)), x] - Simp[n*(n - 1)*(e^2/(e^2*n^2 - 
b^2*c^2*Log[F]^2))   Int[F^(c*(a + b*x))*Sinh[d + e*x]^(n - 2), x], x]) /; 
FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n 
, 1]
 

Maple [A] (verified)

Time = 1.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.11

method	result	size
parallelrisch	\(-\frac {3 F^{c \left (b x +a \right )} \left (\left (c^{2} b^{2} \ln \left (F \right )^{2} e -e^{3}\right ) \cosh \left (3 e x +3 d \right )+\frac {\left (-c^{3} b^{3} \ln \left (F \right )^{3}+e^{2} b c \ln \left (F \right )\right ) \sinh \left (3 e x +3 d \right )}{3}+\left (b c \ln \left (F \right )-3 e \right ) \left (b c \ln \left (F \right )+3 e \right ) \left (b c \ln \left (F \right ) \sinh \left (e x +d \right )-e \cosh \left (e x +d \right )\right )\right )}{4 \left (\ln \left (F \right )^{4} b^{4} c^{4}-10 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}+9 e^{4}\right )}\)	\(147\)
risch	\(\frac {\left (\ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{6 e x +6 d}-3 \ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{4 e x +4 d}-3 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{6 e x +6 d}+3 \ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{2 e x +2 d}+3 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{4 e x +4 d}-\ln \left (F \right ) b c \,e^{2} {\mathrm e}^{6 e x +6 d}-c^{3} b^{3} \ln \left (F \right )^{3}+3 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{2 e x +2 d}+27 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{4 e x +4 d}+3 e^{3} {\mathrm e}^{6 e x +6 d}-3 c^{2} b^{2} \ln \left (F \right )^{2} e -27 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{2 e x +2 d}-27 e^{3} {\mathrm e}^{4 e x +4 d}+e^{2} b c \ln \left (F \right )-27 e^{3} {\mathrm e}^{2 e x +2 d}+3 e^{3}\right ) {\mathrm e}^{-3 e x -3 d} F^{c \left (b x +a \right )}}{8 \left (b c \ln \left (F \right )-e \right ) \left (b c \ln \left (F \right )-3 e \right ) \left (e +b c \ln \left (F \right )\right ) \left (b c \ln \left (F \right )+3 e \right )}\)	\(326\)
orering	\(\frac {4 \ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}-5 e^{2}\right ) F^{c \left (b x +a \right )} \sinh \left (e x +d \right )^{3}}{\ln \left (F \right )^{4} b^{4} c^{4}-10 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}+9 e^{4}}-\frac {2 \left (3 b^{2} c^{2} \ln \left (F \right )^{2}-5 e^{2}\right ) \left (F^{c \left (b x +a \right )} b c \ln \left (F \right ) \sinh \left (e x +d \right )^{3}+3 F^{c \left (b x +a \right )} \sinh \left (e x +d \right )^{2} e \cosh \left (e x +d \right )\right )}{\ln \left (F \right )^{4} b^{4} c^{4}-10 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}+9 e^{4}}+\frac {4 b c \ln \left (F \right ) \left (F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \sinh \left (e x +d \right )^{3}+6 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \sinh \left (e x +d \right )^{2} e \cosh \left (e x +d \right )+6 F^{c \left (b x +a \right )} \sinh \left (e x +d \right ) e^{2} \cosh \left (e x +d \right )^{2}+3 F^{c \left (b x +a \right )} \sinh \left (e x +d \right )^{3} e^{2}\right )}{\ln \left (F \right )^{4} b^{4} c^{4}-10 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}+9 e^{4}}-\frac {F^{c \left (b x +a \right )} b^{3} c^{3} \ln \left (F \right )^{3} \sinh \left (e x +d \right )^{3}+9 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \sinh \left (e x +d \right )^{2} e \cosh \left (e x +d \right )+18 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \sinh \left (e x +d \right ) e^{2} \cosh \left (e x +d \right )^{2}+9 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \sinh \left (e x +d \right )^{3} e^{2}+6 F^{c \left (b x +a \right )} e^{3} \cosh \left (e x +d \right )^{3}+21 F^{c \left (b x +a \right )} \sinh \left (e x +d \right )^{2} e^{3} \cosh \left (e x +d \right )}{\ln \left (F \right )^{4} b^{4} c^{4}-10 \ln \left (F \right )^{2} b^{2} c^{2} e^{2}+9 e^{4}}\)	\(537\)

Input:

int(F^(c*(b*x+a))*sinh(e*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

-3/4*F^(c*(b*x+a))*((c^2*b^2*ln(F)^2*e-e^3)*cosh(3*e*x+3*d)+1/3*(-c^3*b^3* 
ln(F)^3+e^2*b*c*ln(F))*sinh(3*e*x+3*d)+(b*c*ln(F)-3*e)*(b*c*ln(F)+3*e)*(b* 
c*ln(F)*sinh(e*x+d)-e*cosh(e*x+d)))/(ln(F)^4*b^4*c^4-10*ln(F)^2*b^2*c^2*e^ 
2+9*e^4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2228 vs. \(2 (120) = 240\).

Time = 0.18 (sec) , antiderivative size = 2228, normalized size of antiderivative = 16.88 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\text {Too large to display} \] Input:

integrate(F^(c*(b*x+a))*sinh(e*x+d)^3,x, algorithm="fricas")
 

Output:

1/8*((3*e^3*cosh(e*x + d)^6 - 27*e^3*cosh(e*x + d)^4 + (b^3*c^3*log(F)^3 - 
 3*b^2*c^2*e*log(F)^2 - b*c*e^2*log(F) + 3*e^3)*sinh(e*x + d)^6 + 6*(b^3*c 
^3*cosh(e*x + d)*log(F)^3 - 3*b^2*c^2*e*cosh(e*x + d)*log(F)^2 - b*c*e^2*c 
osh(e*x + d)*log(F) + 3*e^3*cosh(e*x + d))*sinh(e*x + d)^5 - 27*e^3*cosh(e 
*x + d)^2 + 3*(15*e^3*cosh(e*x + d)^2 + (5*b^3*c^3*cosh(e*x + d)^2 - b^3*c 
^3)*log(F)^3 - 9*e^3 - (15*b^2*c^2*e*cosh(e*x + d)^2 - b^2*c^2*e)*log(F)^2 
 - (5*b*c*e^2*cosh(e*x + d)^2 - 9*b*c*e^2)*log(F))*sinh(e*x + d)^4 + (b^3* 
c^3*cosh(e*x + d)^6 - 3*b^3*c^3*cosh(e*x + d)^4 + 3*b^3*c^3*cosh(e*x + d)^ 
2 - b^3*c^3)*log(F)^3 + 4*(15*e^3*cosh(e*x + d)^3 - 27*e^3*cosh(e*x + d) + 
 (5*b^3*c^3*cosh(e*x + d)^3 - 3*b^3*c^3*cosh(e*x + d))*log(F)^3 - 3*(5*b^2 
*c^2*e*cosh(e*x + d)^3 - b^2*c^2*e*cosh(e*x + d))*log(F)^2 - (5*b*c*e^2*co 
sh(e*x + d)^3 - 27*b*c*e^2*cosh(e*x + d))*log(F))*sinh(e*x + d)^3 + 3*e^3 
- 3*(b^2*c^2*e*cosh(e*x + d)^6 - b^2*c^2*e*cosh(e*x + d)^4 - b^2*c^2*e*cos 
h(e*x + d)^2 + b^2*c^2*e)*log(F)^2 + 3*(15*e^3*cosh(e*x + d)^4 - 54*e^3*co 
sh(e*x + d)^2 + (5*b^3*c^3*cosh(e*x + d)^4 - 6*b^3*c^3*cosh(e*x + d)^2 + b 
^3*c^3)*log(F)^3 - 9*e^3 - (15*b^2*c^2*e*cosh(e*x + d)^4 - 6*b^2*c^2*e*cos 
h(e*x + d)^2 - b^2*c^2*e)*log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^4 - 54*b*c*e 
^2*cosh(e*x + d)^2 + 9*b*c*e^2)*log(F))*sinh(e*x + d)^2 - (b*c*e^2*cosh(e* 
x + d)^6 - 27*b*c*e^2*cosh(e*x + d)^4 + 27*b*c*e^2*cosh(e*x + d)^2 - b*c*e 
^2)*log(F) + 6*(3*e^3*cosh(e*x + d)^5 - 18*e^3*cosh(e*x + d)^3 - 9*e^3*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (121) = 242\).

Time = 3.30 (sec) , antiderivative size = 1477, normalized size of antiderivative = 11.19 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\text {Too large to display} \] Input:

integrate(F**(c*(b*x+a))*sinh(e*x+d)**3,x)
 

Output:

Piecewise((x*sinh(d)**3, Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x*sinh(d)**3, Eq( 
b, 0) & Eq(e, 0)), (x*sinh(d)**3, Eq(c, 0) & Eq(e, 0)), (-3*F**(a*c + b*c* 
x)*x*sinh(b*c*x*log(F) - d)**3/8 + 3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F) 
- d)**2*cosh(b*c*x*log(F) - d)/8 + 3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F) 
- d)*cosh(b*c*x*log(F) - d)**2/8 - 3*F**(a*c + b*c*x)*x*cosh(b*c*x*log(F) 
- d)**3/8 + F**(a*c + b*c*x)*sinh(b*c*x*log(F) - d)**3/(8*b*c*log(F)) - 3* 
F**(a*c + b*c*x)*sinh(b*c*x*log(F) - d)**2*cosh(b*c*x*log(F) - d)/(4*b*c*l 
og(F)) + 3*F**(a*c + b*c*x)*cosh(b*c*x*log(F) - d)**3/(8*b*c*log(F)), Eq(e 
, -b*c*log(F))), (-F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 - d)**3/8 + 3*F* 
*(a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 - d)**2*cosh(b*c*x*log(F)/3 - d)/8 - 
3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 - d)*cosh(b*c*x*log(F)/3 - d)**2/ 
8 + F**(a*c + b*c*x)*x*cosh(b*c*x*log(F)/3 - d)**3/8 - 9*F**(a*c + b*c*x)* 
sinh(b*c*x*log(F)/3 - d)**3/(8*b*c*log(F)) + 3*F**(a*c + b*c*x)*sinh(b*c*x 
*log(F)/3 - d)**2*cosh(b*c*x*log(F)/3 - d)/(4*b*c*log(F)) - F**(a*c + b*c* 
x)*cosh(b*c*x*log(F)/3 - d)**3/(8*b*c*log(F)), Eq(e, -b*c*log(F)/3)), (F** 
(a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 + d)**3/8 - 3*F**(a*c + b*c*x)*x*sinh( 
b*c*x*log(F)/3 + d)**2*cosh(b*c*x*log(F)/3 + d)/8 + 3*F**(a*c + b*c*x)*x*s 
inh(b*c*x*log(F)/3 + d)*cosh(b*c*x*log(F)/3 + d)**2/8 - F**(a*c + b*c*x)*x 
*cosh(b*c*x*log(F)/3 + d)**3/8 + 9*F**(a*c + b*c*x)*sinh(b*c*x*log(F)/3 + 
d)**3/(8*b*c*log(F)) - 3*F**(a*c + b*c*x)*sinh(b*c*x*log(F)/3 + d)**2*c...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.02 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 3 \, e x + 3 \, d\right )}}{8 \, {\left (b c \log \left (F\right ) + 3 \, e\right )}} - \frac {3 \, F^{a c} e^{\left (b c x \log \left (F\right ) + e x + d\right )}}{8 \, {\left (b c \log \left (F\right ) + e\right )}} + \frac {3 \, F^{a c} e^{\left (b c x \log \left (F\right ) - e x\right )}}{8 \, {\left (b c e^{d} \log \left (F\right ) - e e^{d}\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 3 \, e x\right )}}{8 \, {\left (b c e^{\left (3 \, d\right )} \log \left (F\right ) - 3 \, e e^{\left (3 \, d\right )}\right )}} \] Input:

integrate(F^(c*(b*x+a))*sinh(e*x+d)^3,x, algorithm="maxima")
 

Output:

1/8*F^(a*c)*e^(b*c*x*log(F) + 3*e*x + 3*d)/(b*c*log(F) + 3*e) - 3/8*F^(a*c 
)*e^(b*c*x*log(F) + e*x + d)/(b*c*log(F) + e) + 3/8*F^(a*c)*e^(b*c*x*log(F 
) - e*x)/(b*c*e^d*log(F) - e*e^d) - 1/8*F^(a*c)*e^(b*c*x*log(F) - 3*e*x)/( 
b*c*e^(3*d)*log(F) - 3*e*e^(3*d))
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 1211, normalized size of antiderivative = 9.17 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\text {Too large to display} \] Input:

integrate(F^(c*(b*x+a))*sinh(e*x+d)^3,x, algorithm="giac")
 

Output:

1/4*(2*(b*c*log(abs(F)) + 3*e)*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1 
/2*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(ab 
s(F)) + 3*e)^2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b*c*x*sgn(F) + 1/2* 
pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4 
*(b*c*log(abs(F)) + 3*e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 3*e)* 
x + 3*d) + I*(I*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*s 
gn(F) - 1/2*I*pi*a*c)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) 
 + 48*e) - I*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn 
(F) + 1/2*I*pi*a*c)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b*c*log(abs(F)) 
+ 48*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 3*e)*x + 3*d) - 3/4*(2*(b 
*c*log(abs(F)) + e)*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*s 
gn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + e)^ 
2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/ 
2*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs 
(F)) + e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) + 3*I*(-I* 
e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi 
*a*c)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) + 16*e) + I*e^( 
-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a 
*c)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b*c*log(abs(F)) + 16*e))*e^(a*c* 
log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) + 3/4*(2*(b*c*log(abs(F)) - ...
 

Mupad [B] (verification not implemented)

Time = 3.53 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.26 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=-\frac {F^{a\,c+b\,c\,x}\,\left (-b^3\,c^3\,{\mathrm {sinh}\left (d+e\,x\right )}^3\,{\ln \left (F\right )}^3+3\,b^2\,c^2\,e\,\mathrm {cosh}\left (d+e\,x\right )\,{\mathrm {sinh}\left (d+e\,x\right )}^2\,{\ln \left (F\right )}^2-6\,b\,c\,e^2\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,\mathrm {sinh}\left (d+e\,x\right )\,\ln \left (F\right )+7\,b\,c\,e^2\,{\mathrm {sinh}\left (d+e\,x\right )}^3\,\ln \left (F\right )+6\,e^3\,{\mathrm {cosh}\left (d+e\,x\right )}^3-9\,e^3\,\mathrm {cosh}\left (d+e\,x\right )\,{\mathrm {sinh}\left (d+e\,x\right )}^2\right )}{b^4\,c^4\,{\ln \left (F\right )}^4-10\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+9\,e^4} \] Input:

int(F^(c*(a + b*x))*sinh(d + e*x)^3,x)
 

Output:

-(F^(a*c + b*c*x)*(6*e^3*cosh(d + e*x)^3 - 9*e^3*cosh(d + e*x)*sinh(d + e* 
x)^2 - b^3*c^3*sinh(d + e*x)^3*log(F)^3 + 7*b*c*e^2*sinh(d + e*x)^3*log(F) 
 + 3*b^2*c^2*e*cosh(d + e*x)*sinh(d + e*x)^2*log(F)^2 - 6*b*c*e^2*cosh(d + 
 e*x)^2*sinh(d + e*x)*log(F)))/(9*e^4 + b^4*c^4*log(F)^4 - 10*b^2*c^2*e^2* 
log(F)^2)
 

Reduce [F]

\[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \sinh \left (e x +d \right )^{3}d x \right ) \] Input:

int(F^(c*(b*x+a))*sinh(e*x+d)^3,x)
 

Output:

f**(a*c)*int(f**(b*c*x)*sinh(d + e*x)**3,x)